| L(s) = 1 | − 3-s − 3·5-s − 7-s + 9-s − 3·11-s + 5·13-s + 3·15-s − 17-s − 2·19-s + 21-s − 6·23-s + 4·25-s − 27-s − 6·29-s + 4·31-s + 3·33-s + 3·35-s + 11·37-s − 5·39-s − 12·41-s + 43-s − 3·45-s − 12·47-s + 49-s + 51-s − 9·53-s + 9·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s − 0.377·7-s + 1/3·9-s − 0.904·11-s + 1.38·13-s + 0.774·15-s − 0.242·17-s − 0.458·19-s + 0.218·21-s − 1.25·23-s + 4/5·25-s − 0.192·27-s − 1.11·29-s + 0.718·31-s + 0.522·33-s + 0.507·35-s + 1.80·37-s − 0.800·39-s − 1.87·41-s + 0.152·43-s − 0.447·45-s − 1.75·47-s + 1/7·49-s + 0.140·51-s − 1.23·53-s + 1.21·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5209769691\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5209769691\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 19 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.019507455750696322517359396577, −7.60071241079856865939505009051, −6.52974126119963944803026329454, −6.15891201432989065467850720102, −5.20983774777158442191812465551, −4.35865858694941866044297529040, −3.78371489917505067880314892179, −3.02402476613205357995053379893, −1.73527382204318293609960924592, −0.38832682507999745116601502102,
0.38832682507999745116601502102, 1.73527382204318293609960924592, 3.02402476613205357995053379893, 3.78371489917505067880314892179, 4.35865858694941866044297529040, 5.20983774777158442191812465551, 6.15891201432989065467850720102, 6.52974126119963944803026329454, 7.60071241079856865939505009051, 8.019507455750696322517359396577
